What is an easy way to see that torus with two points removed is homotopy equivalent to wedge of three circles?

I am trying to see it by viewing the torus as a square with two sides identified, but the intuition isn't clear to me, unlike the case with one point removed.

enter image description here

I am trying to see it as each triangle being retracted to its boundary, but I have no idea how that results in wedge of three circles.



1 Answer 1


You can actually see the three circles in this picture! Look at the three edges incident to the bottom-left corner in your picture. These all become circles after we do the identifications, and they meet at a single point (all four vertices of the rectangle become the same point after identifications). And, after identifications, these three edges are the entire space!

  • $\begingroup$ this is a stupid question, but why only two circles if we have one point punctured, instead of three? $\endgroup$
    – Phil
    Jan 19, 2021 at 18:53
  • $\begingroup$ Where would the three circles come from? If you draw the same diagonal line, only one of the triangles will have a hole in it. This triangle will deformation retract onto its boundary, but the other one will not. $\endgroup$ Jan 19, 2021 at 20:39
  • $\begingroup$ @diradeltafunk oh yeah, I see your point. I just have terrible intuititon for these lol. $\endgroup$
    – Phil
    Jan 19, 2021 at 21:36

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